Nabla Seminar 2026 Spring

Grad Analysis Seminar at Rutgers University

Grad Analysis $\nabla$ Seminar is a graduate seminar at Rutgers University, Department of Mathematics, Orgnised by Qi Ma, Aprameya and Anupam since Nov. 2024.

This page is for seminars held in 2026 Spring. This semester we meet regularly on Monday at 4:00 p.m. in the Mathematics Graduate Lounge (Hill 701). We are looking for speakers. Please let me, Aprameya or Anupam know if you want to give a talk.

To see seminars in other semesters, please refer to Nabla.

2026 Apr.20th , Speaker: Kangbai Yan

Title: Singular Initial Value Problems for Wave Equations with Applications to Black Holes

Abstract: This talk concerns the singular initial value problem for a model nonlinear wave equation near a spacelike singularity. I will begin by briefly recalling the standard local existence theory and breakdown criteria for wave equations with regular initial data, together with Fritz John’s classical finite-time blow-up result. Motivated by explicit singular solutions of the Einstein vacuum equations, such as Kasner and Schwarzschild spacetimes, I will then discuss the model equation $\[ \Box_g \phi = (\partial_t \phi)^2 \]$ on a singular background metric (g). The central question is whether one can prescribe asymptotic behavior at the singular time and construct solutions evolving away from it. I will explain the construction of approximate solutions and the accompanying energy estimates for the remainder. Time permitting, I will also comment on the relation of this problem to the study of black hole interiors, in particular to questions surrounding strong cosmic censorship.

2026 Apr.13th , Speaker: Priya Kaveri

Title: The space of stationary m-dimensional surfaces

Abstract: In this talk, I will discuss Brian White’s work on the local structure of the space of immersed m-dimensional surfaces that are stationary for a broad class of elliptic parametric functionals, including the area functional. After reviewing the first and second variation formulas, I will explain how one accounts for the degeneracy caused by reparametrization invariance, leading to a Banach manifold structure on the space of stationary immersions.

2026 Apr.6th , Speaker: Jakub Niksinski

Title: Introduction to Decoupling Inequalities and Vinogradov's Mean Value Theorem

Abstract: Assume we are given a curved surface and a collection of Schwartz functions which are frequency supported on small disjoint neighborhoods of that surface. Decoupling inequalities measure, in terms of Lp norms, how the size of the sum of these functions relates to the sizes of the individual components. These inequalities were introduced in 2000 by Thomas Wolff in his influential work on the local smoothing conjecture for the wave equation. In 2015, Bourgain and Demeter revolutionized the field by introducing techniques allowing one to prove decoupling in optimal ranges of exponents. Since then, decoupling inequalities and the techniques surrounding them have become a central tool in Euclidean harmonic analysis with far-reaching applications to: restriction problems, PDEs, incidence geometry, geometric measure theory (specifically Kakeya-type problems), and even analytic number theory. We will briefly go over some of these applications and see how Vinogradov's Mean Value Theorem easily follows from the general decoupling inequality for the moment curve proved by Bourgain, Demeter, and Guth in 2016. I intend to spend most of the talk giving a sketch of the proof of the decoupling inequality for the parabola and showcasing some of the crucial ideas involved.

2026 Mar.30th , Speaker: Owen Drummond

Title: The Alt-Caffarelli Free Boundary Problem with a Volume Constraint

Abstract: Caffarelli and Alt published their landmark paper on the Bernoulli problem in 1981. Concurrently, they had a parallel problem in mind: what if we prescribe the volume of the positivity set? As it turns out, classical methods fail to provide minimizers for the most straightforward unconstrained functional. This necessitates a more sophisticated penalized approach to recover the powerful techniques used in the original program. Instead, we require some heavier analysis, estimates, a different kind of penalized functional to apply these methods. Remarkably, all of the same regularity results for the minimizer carry over for the volume constraint problem, and the volume of the positivity set for the penalized minimizer automatically corrects to the prescribed volume in the problem.

2026 Mar.23rd , Speaker: Sam Wallace

Title:

Abstract: Thin elastic structures (TES), whether 1D rods or 2D sheets, are becoming widespread in mechanical design, used to combat heart disease and explore the depths of space. The basic reason for their use is that TESs deform by local isometry, which allows for any length shortening map of a TES to be nearly obtained (Nash-Kuiper). Thus many qualitatively different shape changes can be obtained by compression. One type of shape change, origami, is becoming a widespread design paradigm; it is characterized by flat, rigid panels with flexible joints. I will present 2-3 results that illuminate fine properties of maps modeling flat-folded origami. This work is part of my thesis.

2026 Mar.2nd , Speaker: Aprameya Girish Hebbar

Title: Metric Flows: Heat Kernels, F-convergence and Compactness

Abstract: In his landmark 2020 work, R. Bamler introduced metric flows which are weak synthetic analogues of smooth flows. Roughly, a metric flow consists of time-slices of metric spaces equipped with heat kernel type measures, providing a "parabolic analogue of a metric space". I will explain how the axioms of metric flows arise naturally from smooth Ricci flow, the convergence notion (“F-convergence”), and if time permits, how this yields a compactness theory for sequences of (super) Ricci flows.

2026 Feb.16th , Speaker: Qi Ma

Title: About Liouville’s Theorem – From harmonic function to nonlinear PDEs

Abstract: Liouville’s Theorem plays a fundamental role in the uniqueness of solutions. In this talk, we will start from the most trivial case – harmonic function. Classical proof, proof by energy estimation and proof by Fourier transform will be presented. Finally, we will generalize our result to nonlinear equations and introduce the some results of removable singularities in Navier-Stokes.

2026 Feb.9th , Speaker: Sammy Thiagarajan

Title: Schauder Estimates three different ways.

Abstract: Schauder estimates are classical estimates for second order, uniformly elliptic PDEs. Naturally, there are lots of ways to prove them! We will discuss three interesting ways in this talk.

2026 Feb.2nd , Speaker: Junyoung Park

Title: On Eells and Sampson theorem

Abstract: The idea of using heat type equations to solve problems in geometry has been proven to be an extremely powerful method. In this talk, we will discuss one of the earliest success of such approach, which is the use of harmonic map heat flow to find minimizing harmonic maps in a given homotopy class, under a geometric constraint on the target manifold.